发表机构
School of Mathematics and Statistics, Guizhou University; Department of Mathematics, The University of Hong Kong（贵州大学数学与统计学院; 香港大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了超插值$L^2$收敛充要条件的开放问题，证明$L^1$-$L^2$ MZ条件结合多项式渐近逼近性质是答案，还明确了求积与超插值收敛条件的层级关系。
AI 中文摘要
自1995年Sloan提出超插值以来，确定其$L^2$收敛的充要条件一直是未解决的问题。我们证明$L^1$-$L^2$ Marcinkiewicz-Zygmund（MZ）条件结合多项式的渐近函数逼近性质就是答案。我们进一步证明最优$L^1$-$L^2$ MZ常数与超插值算子的算子范数一致，且具有自然的巴拿赫空间对偶解释。通过显式构造，我们还证明求积收敛的波利亚经典条件不足以保证超插值的$L^2$收敛，这揭示了线性泛函（求积公式）收敛与线性算子（超插值算子）收敛之间的根本区别。我们建立了支配求积与超插值收敛的稳定性和精度条件的严格逻辑层级。
英文摘要
It has remained open to identify the necessary and sufficient conditions for the $L^2$ convergence of hyperinterpolation since it was introduced by Sloan in 1995. We show that the $L^1$-$L^2$ Marcinkiewicz-Zygmund (MZ) condition, together with the asymptotic functional approximation property for polynomials, is the answer. We further prove that the optimal $L^1$-$L^2$ MZ constant coincides with the operator norm of the hyperinterpolation operator, and it admits a natural Banach space duality interpretation. With an explicit construction, we also show that Pólya's classical conditions for quadrature convergence are not sufficient for the $L^2$ convergence of hyperinterpolation. This reveals a fundamental distinction between the convergence of linear functionals (quadrature formulas) and that of linear operators (hyperinterpolation operators). We establish a strict logical hierarchy for the stability and accuracy conditions governing the convergence of quadrature and hyperinterpolation.
Comments24 pages, 3 figures
